Bovine Bridge Battle

Each of Farmer John's $N$ cows ($4 \le N \le 1000$) is waiting in the main pasture, with cow $i$ standing at the integer coordinates $(X_i, Y_i)$ ($-10^9 \le X_i, Y_i \le 10^9$).

The cows want to form groups of four to play Bridge, their new favorite card game. Each group must satisfy the following constraint: four cows may team up if and only if there exists some point $P$ in the plane (not coincident with any of the four cows' positions) such that rotating each cow of the group $180^\circ$ about $P$ lands exactly on the position of another cow in the same group.

In other words, the four positions must be point-symmetric about some center $P$. Determine how many sets of four cows can form a Bridge group.

For example, suppose eight cows stand at the following eight points.

                  |
                 f*
                  |             a = (-3, 1)    e = (-1, 1)
           b*     |             b = (-2, 2)    f = ( 0, 3)
        a      e  |             c = (-3, 0)    g = ( 2, 0)
         *     *  |             d = (-2, 0)    h = ( 3, 0)
         c  d     |     g  h
---------*--*-----+-----*--*---------
                  |

Then there are exactly three legal groups: $\{a, b, e, d\}$ (rotating about $(-2, 1)$), $\{b, c, e, f\}$ (about $(-1.5, 1.5)$), and $\{c, d, g, h\}$ (about $(0, 0)$).

All given cow positions are distinct and are provided in no particular order. The answer is guaranteed to fit in a signed 32-bit integer.

• Line 1: A single integer $N$.
• Lines 2 to $N+1$: Line $i+1$ contains two space-separated integers $X_i$ and $Y_i$.

• Line 1: A single integer — the number of sets of four cows that can form a valid Bridge group.